Bevel tool creating intersecting faces
The first thing I would do is delete all the faces on the top along with those nasty triangles and poles... tris and ngons do not subdivide well with bevels.

Try to keep your geometry with quads.



Once you delete the offending faces, select the edge loops you want to bevel, and on the top view extrude and scale. The idea is to create a new set of edges and faces that follow the topology using quads.



Those edges will bevel nicely now...



If you need to bevel the outer edge loops do the same.





After that then just fill the gaps with the best topology you can come up with.